A novel approach to assessing natural resource injury with Bayesian networks

Quantifying the effects of environmental stressors on natural resources is problematic because of complex interactions among environmental factors that influence endpoints of interest. This complexity, coupled with data limitations, propagates uncertainty that can make it difficult to causally associate specific environmental stressors with injury endpoints. The Natural Resource Damage Assessment and Restoration (NRDAR) regulations under the Comprehensive Environmental Response, Compensation, and Liability Act and Oil Pollution Act aim to restore natural resources injured by oil spills and hazardous substances released into the environment; exploration of alternative statistical methods to evaluate effects could help address NRDAR legal claims. Bayesian networks (BNs) are statistical tools that can be used to estimate the influence and interrelatedness of abiotic and biotic environmental variables on environmental endpoints of interest. We investigated the application of a BN for injury assessment using a hypothetical case study by simulating data of acid mine drainage (AMD) affecting a fictional stream‐dwelling bird species. We compared the BN‐generated probability estimates for injury with a more traditional approach using toxicity thresholds for water and sediment chemistry. Bayesian networks offered several distinct advantages over traditional approaches, including formalizing the use of expert knowledge, probabilistic estimates of injury using intermediate direct and indirect effects, and the incorporation of a more nuanced and ecologically relevant representation of effects. Given the potential that BNs have for natural resource injury assessment, more research and field‐based application are needed to determine their efficacy in NRDAR. We expect the resulting methods will be of interest to many US federal, state, and tribal programs devoted to the evaluation, mitigation, remediation, and/or restoration of natural resources injured by releases or spills of contaminants. Integr Environ Assess Manag 2024;20:562–573. Published 2023. This article is a U.S. Government work and is in the public domain in the USA. Integrated Environmental Assessment and Management published by Wiley Periodicals LLC on behalf of Society of Environmental Toxicology & Chemistry (SETAC).


INTRODUCTION
Quantifying the effects of environmental stressors on natural resources is problematic because of the complexity of interrelated environmental factors within ecosystems that influence endpoints of interest (e.g., population size).This complexity propagates uncertainty that can make it challenging to causally associate specific stressors, such as contaminants, with effects on environmental endpoints such as specific species or their habitats.Approaches to quantifying and modeling environmental stressors have traditionally included frequentist statistical methods such as analysis of variance, generalized linear models, and structural equation models.However, recent attention on the misuse of frequentist p-values in research (Wasserstein & Lazar, 2016), including in aquatic toxicology (Erickson & Rattner, 2020), has motivated the need for alternative statistical approaches, particularly those that can explicitly track propagation of uncertainty.
Natural resource injury, as defined in the Natural Resource Damage Assessment and Restoration (NRDAR) regulations, is a "measurable adverse change, either long-or short-term, in the chemical or physical quality or the viability of a natural resource resulting either directly or indirectly from exposure to a discharge of oil or release of a hazardous substance" (43 CFR §11.14(v)).Injury encompasses the phrases "injury," "destruction," and "loss" of surface water, ground water, air, geologic, and biological resources (43 CFR §11.64).The objectives of NRDAR for state, tribal, and federal agencies are to assess natural resources injured by the releases of oil or other hazardous substances and to restore the lost natural resource services.This happens through a complex process of establishing a hazardous substance exposure, quantifying the magnitude of the injury to the natural resource, and establishing remediation and restoration plans to bring the resource back to baseline and/or reference conditions (Gouguet et al., 2009;Rohr et al., 2013;Sanders et al., 2016).To assess injury, trustees from state, tribal, and/or federal agencies form case teams that include scientists, economists, and solicitors to conduct a damage assessment for individual sites or incidents (i.e., a NRDAR case).Integral to this process is technical scientific support to identify and measure injury to natural resources for economists and solicitors to determine the monetary or project-based natural resource claims for restoration (Kraus et al., 2023); however, assessment of injury to the environment can be challenging.Common issues that affect the ability of researchers to demonstrate natural resource injury include: (1) limited evidentiary data; (2) difficulty characterizing contaminant fate and effects, including contaminant mixtures; (3) confusing and complicated interpretation of statistically measured outcomes; (4) difficulty defining the temporal and spatial scales; and (5) lack of baseline data.In that context, probabilistic statistical tools are particularly well-suited for injury assessment.
Bayesian networks (BNs) are probabilistic statistical tools (Jensen & Nielsen, 2007) that have been used in the field of natural resources for several decades to determine the influence of both abiotic and biotic environmental variables, and the interrelatedness of these variables on environmental endpoints of interest (Aguilera et al., 2011;Marcot, 2017;Marcot et al., 2006).Advantages of BNs over other statistical approaches are that they express variation and uncertainty in parameters, identify key environmental variables that most influence endpoints of study, and depict contaminanteffect causal relationships.Importantly, BNs can be used for predictive modeling.With the inclusion of explicit decision and utility variables, BNs can be used in injury assessment as decision-advisory tools for degraded ecosystems affected by anthropogenic stressors.These applications include ecological and toxicological risk assessments (Carriger & Barron, 2020;Moe, Carriger, et al., 2021).
Bayesian networks are structured as variables ("nodes") linked by logical or causal relationships.Bayesian networks are referred to as directed acyclic graphs: directed meaning that the links among nodes are arrows that denote immediate influences or effects, acyclic meaning that BNs do not explicitly include feedback loops (although structuring feedbacks is possible by linking replicates of a BN), and graph meaning a general form of structuring variables in a figure with nodes and links.Links in BNs are sometimes referred to as edges, and nodes leading directly into other nodes are parents of their child nodes.Child nodes are typically represented by conditional probability tables (CPTs) that denote potential outcome states given the combinations of the states in the parent nodes.Nodes with no parents are typically viewed as inputs to the network and are denoted with unconditional (prior) probability values for their states.Nodes typically contain discrete states each with probability values calculated using Bayes' Theorem (Koski & Noble, 2011), and they can also contain equations, constants, and logical functions.
Further, BNs can include explicit decision nodes, such as with alternative management choices, and utility nodes that can denote costs, benefits, and other outcome units that can be linked to any node in the network.Bayesian networks can be structured, and CPTs parameterized, based entirely on a dataset (i.e., case file in BN model terminology) with known variable state values and outcomes even with some missing data, or based on expert knowledge, or in combination.They can also be useful decision-advisory tools in the context of structured decision-making.In sum, although BNs are but one tool in the modeling toolbox, they are a remarkably flexible construct for addressing a wide array of problem types and issues.More pointedly, BNs can incorporate variation in variables and explicitly demonstrate the implications of that variation on output conditions by propagating uncertainty throughout a network.Moreover, causal network structures, including BNs, are increasingly used to infer environmental impacts from experimental and observational studies (Arif & MacNeil, 2022).
Although BNs have been applied in ecotoxicology previously (Landis, 2021;Moe, Wolf, et al., 2021), the application of BNs in injury assessment for NRDAR cases is novel.
Here we present a proof-of-concept example of how BNs may be helpful in addressing many of the issues associated with establishing natural resource injury (Table 1).Figure 1 illustrates a conceptual model framework that represents the basis for developing a causal web to determine the probability of injury.For most NRDAR cases, contaminants in different abiotic and biotic media (e.g., sediment, water, in living organisms) are measured among different locations in the area of interest.The fate of contaminants refers to the media in which contaminants occur and to which they can be transported, whereas the effect of contaminants is the lethal or sublethal response of living organisms to exposure.Environmental injury is a direct or indirect response to that exposure.For the purposes of demonstrating the utility of BNs in natural resource injury assessment, we use published data and expert knowledge to establish an example injury determination such as might be developed in a NRDAR case using the conceptual model (Figure 1A).Ascertaining injury depends on the site-specific impacts of any contaminant(s), the effect they have on target organisms (e.g., altered behavior, suppressed immune function, death), and the contaminant fate (e.g., how the contaminant accumulates in sediment, water column, and biota) on whatever the outcome of interest is for the injury (e.g., loss of individuals in local animal populations and loss of biomass).The primary objectives for building this BN were to: (1) demonstrate how to construct an effective BN to determine the probability of natural resource injury, (2) provide key considerations for how to structure models and leverage data sources, (3) demonstrate interpretation of different BN results based on varying model inputs and outputs, and (4) discuss the practicality of further developing such BNs as injury assessment and decision-support tools in NRDAR cases.

METHODS
We developed the BN model presented here in Netica (v.6.09; Norsys Software Corporation) as a demonstration of procedures and potential application.This demonstration model pertains to calculating potential injury of a fictional federally endangered bird species with the pseudonym "Pine Dipper" (nom de guerre Cinclus fakeus), characterized as being closely associated with stream and riparian con-   (Darwiche, 2009;Hobbs & Hooten, 2015;Marcot, 2006Marcot, , 2017)).Abbreviation: DOC, dissolved organic carbon.
FIGURE 1 (A) conceptual model framework used to develop a causal web for assessing injury to a fictional, stream-dwelling bird (Pine Dipper), and (B) map of case example with acid mine drainage (AMD) affected stream in the Assessment Area (AA).Twelve locations were evaluated in the Assessment Area (AA1-12) and three reference locations (Ref1-3) that could not be affected by the former mine were included in our example abandoned mine contamination on stream water pH levels and concentration of copper (Cu) in the stream sediments.
We demonstrate how a BN can be structured and parameterized using a combination of data and expert knowledge.Note that detailed guides for building BNs exist elsewhere (Heckerman, 2008;Jensen, 1996;Jensen & Nielsen, 2007;Kjaerulff & Madsen, 2007;Woodberry & Mascaro, 2014), but our objective here is a proof-of-concept for the application of BNs to help with injury determination in the NRDAR process.

Simulated case
We simulated data for a fictional NRDAR (i.e., realistic but not reflective of any specific case) case involving a contaminated mining site in the mountainous western United States, similar to actual sites (Clements et al., 2010;Kotalik et al., 2023).In this example, sulfur mining occurred for several years before the mine was reopened as an open-pit Cu mine.The site closed a decade ago, but persistent acid mine drainage (AMD) is related to ongoing impacts including high acidity (stream pH ~2 commonly observed immediately below the AMD source and elevated concentrations of Cu in water and sediments).The affected stream bisects the mine and flows an additional 5.6 km to the confluence of Tributary Creek and 8 km further downstream to the confluence of Big River (Figure 1B).
We imposed three nearby reference sites (Ref1-3; green triangles; Figure 1B) and 12 sampling locations in the AMD source Assessment Area (Assessment Area; orange triangles) of varying distances from the AMD source (yellow square; Figure 1B), simulating the spatial sampling regime that investigators might use for this contaminated system.Reference sites were upstream of the AMD source (Ref1) and upstream of the confluences with Tributary Creek (Ref2) and Big River (Ref3).The sampling locations in the Assessment Area (AA1-12) were distributed approximately every 1-2 km downstream from the AMD source, with the most downstream site below the confluence with the Upper Big River (AA12).To determine injury to populations (i.e., size of the local population in the NRDAR assessment area) of the fictional Pine Dipper found along several stretches of this watershed, we simulated water pH and sediment Cu values for all sampling locations.Pine Dippers are assumed to be effective indicators of stream quality because they depend on aquatic habitat and prey resources (aquatic invertebrates) in this watershed.Notably, estimates of Pine Dipper population size are simplified as number of birds per reach here, but more sophisticated population modeling would likely be needed for population estimates using "realworld" NRDAR cases.

Data simulation
To model sediment Cu concentrations at each sampling location, we fit an exponential decay model based on proximity to the AMD source and parameterized the model based on actual values previously reported (Covelli et al., 2001): where sediment Cu at site i (AA1-12) was determined by Cu 0 , the sediment Cu adjacent to mine site; k is the decay coefficient, set at k = 0.198 reported by Covelli et al. (2001); and distance is the distance from the AMD source in stream km.Sediment Cu has been measured as high as 1000-4000 mg/kg (Damous et al., 2002;Rygg, 1985), and we set our intercept at 400 mg/kg dw sediment Cu.
Water pH was modeled using a logistic growth curve, where water pH at site i is predicted by pH max = 7.5, which represents the maximum expected average pH in the stream under baseline conditions; pH 0 = 2.5 based on the scenario indicating that pH was routinely near 2 directly below the AMD source inputs; c is the coefficient of growth, set at 0.2; and distance is the distance from the AMD source in stream km.For each location's simulated Cu and pH value, we generated random error (Table 2) to represent stochasticity that is often encountered with field-collected data.We used the "rnorm" function in R (v. 4.2.1)statistical computing software (R Core Team, 2022), with Gaussian probability distributions generated for each site assuming mean chemistry values equal to the site-assigned chemistry values (i.e., Cu or pH) and standard deviations that approximated the known variability of these water chemistry values at AMD source Assessment Area.Based on the respective probability distributions, we generated 10 random values (i.e., replicates) for Cu and pH for each site (Supporting Information: Table S5).We chose 10 sample replicates for each location to reflect a realistic number of samples collected for a case to account for within-site variability in stream chemistry (Supporting Information: Table S1).

Bayesian network model development
We developed the BN model from our simulated data, literature reviews, and our collective knowledge of bird ecology and stream ecotoxicology, following four main steps.First, we identified the key variables potentially associated with the contaminant loads and how they might influence the local Pine Dipper population.Second, we linked the variables into a causal network to denote the major pathways in which these influences could play out (Figure 2).Third, we then identified how each variable in the network can be represented by specific units of measure and by values of state ranges; this step equates in BN modeling to discretizing continuous-value nodes, that is, defining discrete, exclusive state ranges for each node (Supporting Information: Table S1).Fourth, we parameterized all child nodes in the network by denoting their conditional probability values or conditional equations, in part, from our collective knowledge and by incorporating the simulated dataset using a machine-learning algorithm.The modeled average values for each site's water pH and sediment copper (Cu; mg/kg dry weight) were calculated using Equations ( 1) and ( 2), and the standard deviation (SD) of each was used to model random error around the average for each of the 10 replicates.
Such algorithms can include various methods, such as our use of expectation maximization (Do & Batzoglou, 2008), which is a convergent log-likelihood function that sets conditional probability values to best match patterns in a database with values of variables including known outcome conditions.We used expert knowledge obtained from the literature and combined that with sediment Cu and water pH data to identify key causal relationships by which to structure a BN depicting the potential impacts of the AMD source on the Pine Dipper (Figure 2) in the Assessment Area.This expert knowledge included an extensive literature review to populate the CPTs and to specify the states in the nodes (Supporting Information: Table S1).For example, two pertinent studies explored how water pH affects Cinclidae clutch size (Ormerod et al., 1991) and sediment Cu toxicity impacts macroinvertebrates (MacDonald et al., 2000).For demonstration purposes, we restricted our water chemistry simulation to Cu and pH, which we used to inform our effect concentration benchmarks.However, estimates of metal toxicity using bioavailability-based models that integrate ancillary water chemistry parameters that are known to modify metal toxicity in the field would likely be needed for actual NRDAR cases.

Bayesian network model structure
The overall structure of our BN model tracks how low values of stream water pH and high concentrations of sediment Cu may negatively affect the local Pine Dipper population (see Supporting Information: Table S1 for   in low invertebrate community biomass, the Pine Dipper's primary food resource, which lowers adult Pine Dipper annual survival rates.Higher levels of sediment Cu concentration result in low invertebrate community biomass, as well as higher levels of Cu concentration in invertebrate tissues.These translate to higher Cu concentrations in Pine Dipper livers because invertebrates are a key component of Pine Dipper diets.Higher Cu concentrations in Pine Dipper tissue also cause lower adult Pine Dipper annual survival.Low Pine Dipper nest productivity and low adult annual survival then compound to a loss of density in the Assessment Area.Finally, Pine Dipper density in the Assessment Area is compared with density at the reference sites to estimate injury.The reference sites' Pine Dipper density node is obtained from the outcome node from a parallel analysis that provides values for that node (Supporting Information: Figure S1); as such, it is characterized as a response/output node that is then used as an input node for the overall injury evaluation.
The model is structured so that it can be run with data on the input variables of water pH and sediment Cu concentration from individual sample locations in the field, resulting in site-specific projections of potential Pine Dipper density losses relative to reference site conditions.Then, all sitespecific projections can be summarized statistically as mean and standard deviation to denote overall losses and variability across the sample locations.The reduction in adult annual survival, nest productivity, and density are all endpoints that can be used by economists to quantify injury in NRDAR cases (Baker et al., 2020; Damage Assessment and Restoration Program (U.S.), 2006; Desvousges et al., 2018).

Bayesian network model testing
We conducted a sensitivity analysis of the model structure by analyzing the degree to which the Assessment Area Pine Dipper population node is sensitive to all other nodes contributing to that outcome in the model.Sensitivity analysis can be useful for identifying nodes with greatest influence, and thus, which variables might be most important to measuring and determining accurate values, such as, in our demonstration BN, water pH and sediment Cu concentration.Furthermore, NRDAR case teams may use the results of the sensitivity analyses to focus negotiations and potential additional assessments.
An empirical dataset, if available, could also be used to conduct model calibration accuracy analysis and to parameterize probability values for parts of the model.It is important to note, however, that calibration accuracy refers to how well a model conforms to a set of data used, at least in part, to structure and parameterize the model.Calibration accuracy is not validation accuracy, which entails testing a structured model against an independent dataset not used in its construction and parameterization.It may be rare in ecotoxicology modeling to acquire such independent datasets or to have enough field samples by which to adequately conduct cross-validation.For these reasons, we suggest that independent peer review could be sought on a general BN structure and then also on an operational BN model fully parameterized with probability values (Marcot et al., 2006).

RESULTS
The Pine Dipper population injury BN describes a causal relationship of how sediment Cu and water pH affected our species of interest (Figure 2).The effects of these chemical stressors on Pine Dipper populations were propagated through direct and indirect effect on their prey resources (i.e., stream invertebrates), reproduction, and annual survival.Results of the sensitivity analysis unsurprisingly revealed that injury to Pine Dipper was most affected by local population responses resulting from variation in clutch size and annual adult survival (Supporting Information: Table S2).However, water pH had more of an influence on Pine Dipper injury than sediment Cu in this simulated example.Reference sites were calculated separately and used as an input node in the model to compare with the predicted population response of dippers to contamination (Supporting Information: Figure S1).Traditional comparisons limited to water pH and sediment Cu data for sampling locations versus distance from the AMD source were correlative and lacked quantitative inference on probability of population effects (Figure 3).Clutch sizes in real-world Cinclidae declined with pH lower than 4 (Ormerod et al., 1991), and four of our hypothetical test sites (AA1-4; Figure 3A) fell below this threshold.This will likely result in lower clutch sizes at these assessment sites, whereas pH at all reference sites is far greater than this threshold (Figure 3B).MacDonald et al. (2000) established threshold effect concentrations (TEC) and probable effect concentrations (PEC) for sediment metals and macroinvertebrate survival.Four assessment sites were above the PEC (AA1-4; Figure 3C), and an additional four were above the TEC (AA5-8).Approximately half the test sites had sediment Cu concentrations above the TEC (Figure 3C).
The BN predicted the probability of local Pine Dipper population injury (Figure 4) by using a submodel of the reference site data as a baseline for comparison.The conclusions from the BN model were similar to conclusions reached from the prior analysis using effect thresholds for pH and sediment Cu concentrations (Figure 3); Pine Dippers are still likely to be injured at sites AA1-4; however, we now have additional information on the probability of injury (Figure 4).
Essentially, the BN is a decision-advisory tool to help understand potential causal mechanisms, such as for local Pine Dipper population decline and for which additional data are needed to reduce the uncertainty of outcomes.Furthermore, because BNs are probabilistic, they can provide more information than a single threshold value such as "pH lower than 4 decreases clutch sizes"; they can be used to calculate the probability of injury related to distance from the AMD source.For example, data collected on site at varying distances from the AMD source (Figure 3) could be used by NRDAR case teams to conclude 90% probability of  injury within the first 2 km downstream from the AMD source, and 10% probability of injury to Pine Dipper populations up to 14 km downstream.The model indicated that 90% injury was probably 1 km downstream and probability decreased to 10% injury 14 km downstream (Figure 4).

DISCUSSION
Here we demonstrate how BNs can be developed and applied in NRDAR injury assessment.We used commonly collected field data (e.g., sediment metal concentrations, water pH), and paired these data with expert knowledge and published data sources to define the BN's nodes, their states, and to parameterize the network's CPTs.A primary objective for constructing this BN was to demonstrate the flexibility in data sources and model structure that can be tailored to case-specific injury assessments, which has great potential to address some of the issues frequently encountered with NRDAR injury assessment (Table 1).The central challenge is teasing out the impacts from the hazardous substance(s) in the unpermitted release from the other factors that influence natural resources.Below, we discuss the efficacy of using causal webs as a decision-advisory tool for assessing injury in NRDAR cases.These scientific outputs can then be used as inputs for economic models to assign dollar values to injury using methods such as habitat equivalency analysis (HEA) and resource equivalency analysis (REA; Baker et al., 2020).Designing relevant and costeffective studies in NRDAR cases-especially those that consider indirect food web effects of contaminants-is sorely needed (Kraus et al., 2023).
The strength of the BN approach is twofold.First, NRDAR case teams can use expert knowledge gleaned from expert elicitation, published literature, or both, to establish how the injury occurred.For example, for Cu in sediment, we used expert knowledge and published literature to construct a causal web demonstrating that high Cu reduced macroinvertebrate biomass and increased Cu bioaccumulation in the Pine Dipper and that lower food quantity and quality affected local population size (Figure 2).Second, the probabilistic approach provides a means by which uncertainty and strength of knowledge can be clearly represented and reveal how outcomes are affected.Bayesian networks can be used to generate testable hypotheses of effects of restoration activities and can help develop priorities for such activities.For example, a BN model could highlight how different restoration activities may be more or less effective at restoring sediment or water quality or estimate expected improvements in local Pine Dipper populations by measuring population densities relative to the restoration action(s) taken.We demonstrate the difference between a traditional threshold approach versus a BN in a NRDAR injury assessment scenario where case teams had water pH and sediment Cu data.Bayesian networks provide greater clarity, quantify probability outcomes, and provide an analysis structure that can be readily updated with new information and data, which are but some of the advantages over "traditional" methods.Using traditional benchmarks, a case team might use PEC thresholds for sediment Cu on macroinvertebrate survival (MacDonald et al., 2000) and water pH on Cinclidae fecundity (Ormerod et al., 1991).Four sites (AA1-4; Figure 3) are greater than these thresholds, and the case team may reasonably conclude possible evidence of injury to macroinvertebrates and the Pine Dipper.The next step would likely involve transporting fieldcollected sediment or water into the laboratory for acute and/or chronic toxicity assessments using surrogate test organisms (e.g., Daphnia, midges, amphipods).These laboratory assessments would serve as proxies for effects of water or sediment contaminants on macroinvertebrate survival and growth.Using field-based and laboratory-derived effect threshold approaches-particularly as independent lines of evidence-makes it difficult to assess injury to the local Pine Dipper population size, the primary injury assessment endpoint of interest.However, BNs can leverage a combination of data sources and published relationships among ecosystem variables to construct a causal web that estimates the probability of Pine Dipper injury as a function of both sediment Cu and water pH and as a function of distance downstream from the AMD source (Figure 4).Furthermore, sensitivity analysis provides information regarding the relative influences of Cu and pH on injury when they are modeled together.A Bayesian network combines as well as identifies the pathways driving the probabilities of injury to help NRDAR practitioners prioritize projects to achieve restoration.Furthermore, formalizing the use of literature and expert knowledge to build BNs could save time and money by reducing the number of site-specific toxicity experiments and field samples collected.
By providing probabilities of outcomes, a BN approach fits well in an injury assessment framework.The injury to the ecosystem in this example reduced local Pine Dipper population size relative to reference conditions and gives us the probability of an adverse outcome.Natural Resource Damage Assessment and Restoration case teams or attorneys may use BNs to advise their project decisions based on their degree of acceptance of injury probability.For example, if a decision-maker is willing to accept a 90% probability of injury as tolerable, we could reasonably state the local population injury from AMD contamination pertained to approximately 1.5 km of stream (Figure 4).Or if the local Pine Dipper population in the Assessment Area has high societal or ecological importance, a decision-maker could assign a lower, more case-specific threshold, such as a 70% probability of injury, suggesting 4.2 km of stream pertained to high population injury.Additionally, BNs can determine which nodes contribute the most and least to local population injury (i.e., sensitivity analysis; Supporting Information: Table S2).For example, in our network, water pH, invertebrate community biomass, and Pine Dipper annual survival had the greatest influences on local Pine Dipper population injury (Supporting Information: Table S2), suggesting that remediation and restoration actions that would improve these metrics would have the greatest effect on reducing injury.In more complicated systems with more Integr Environ Assess Manag 2024:562-573 Published 2023 wileyonlinelibrary.com/journal/ieamunknowns, sensitivity and influence analysis could be useful in identifying parts of the causal network needing more information or research (Marcot, 2012).In general, BNs can be used to couch outcomes as probabilities and to help advise decisions based on the implications of uncertainty.The causal approach, the explicit depiction of uncertainty, and model sensitivity analysis results are powerful tools beyond a PEC and/or TEC approach for those seeking natural resource injury determination.
The most critical caution on the use of BNs is that they are decision-advisory tools, and they should not be used for final decision-making of injury in NRDAR.They can be useful to describe relationships, to identify key environmental factors contributing to potential injury outcomes, and to help advise decisions.They are not deterministic.Outputs from BNs are probabilistic and give the most likely state based on the information in the model, but exact and/or deterministic conclusions such as "reducing Cu by 100 mg/kg will result precisely in a population density of 5.6 Pine Dippers/stream km" would be inappropriate because the output depends on the causal structure and CPTs of the model.Rather, a BN output should be interpreted probabilistically, as discussed above.In addition, BNs that are constructed in the process of injury determination may suggest high probability of injury to a hypothetical endpoint of interest for which field data collection was limited (as in our demonstration).In this case, targeted data collection for the endpoint of interest (e.g., macroinvertebrate biomass and dipper population surveys) would be needed as another line of evidence to aid in injury determination.
Objectives must be well-defined at the start of BN model building, and probability outcomes obtained from the model must be clearly interpreted.A clear, concise statement of objective can help avoid the common issue of objective creep, whereby unclear initial objectives lead to poorly designed models the end user wants to use simultaneously for multiple and potentially incompatible objectives, such as diagnosis, prediction, forecasting, and scenario comparison (Marcot, 2017).A BN that is wellconstructed with clear initial objectives can help predict the probability of a response outcome based on the causal network, and where the incorporation and propagation of uncertainty in the models can be highly useful for aiding in decision-making.For optimal performance, the model structure, including the selected variables and their connections, should be peer-reviewed, and individual variables in causal relationships should be clearly defined so as to be empirically testable.Additionally, the parametrizations for each node must be clearly documented and available for all involved in the injury decision-making process (Landis, 2021;Marcot, 2017).Diagnostic methods such as sensitivity analysis and influence analysis can help identify nodes with the least or greatest influence on the response outcome and can aid understanding the network structure and guiding model updating.Where possible, validating BNs against independent data can be used to test model accuracy and, when compared with model calibration results, can identify degrees of model overfitting (Marcot, 2017).
Bayesian networks are a relatively new way of thinking in the disciplines of ecotoxicology and resource management.Instead of point values or formulaic relationships, a BN uses probability distributions of possible values over each variable, allowing users to estimate uncertainty associated with the predictions (Fenton & Neil, 2012).Uncertainty in a BN model, depicted by probability distributions, can represent frequency distributions of conditions as known from empirical data or as expected from expert knowledge.Bayesian networks then can clearly demonstrate the propagation of such uncertainties and distributions throughout a causal network and the implications for output predictions.Unlike frequentist statistical constructs, BNs can be constructed and fit even with moderate degrees of missing data (Ramazi et al., 2021).From a frequentist approach, ideally 20 observations per variable are needed for adequate fit (Harrell, 2001).However, from a BN approach, one needs enough data to fill in variable levels to induce model structure and can augment areas with limited data using literature values or expert knowledge.To improve model performance, BNs can then be updated with more information as it is gathered.
In this article, we demonstrated the use of BNs as a decision-advisory tool in the assessment of NRDAR injury and provide key considerations for the construction, structure, and interpretation of BN results.We simulated a relatively simplified, but reasonable, case study for demonstration.However, many NRDAR cases are much more complex, with interactions among many stressors, in varying environmental media, and affecting numerous species.This complexity can be a major challenge in injury assessment, but BNs can provide measures of uncertainty propagated from these confounding factors to aid practitioners in the assessment of injury and to guide future assessment efforts.Bayesian networks can account for unobserved influences in the system (i.e., latent variables) by broadening the probability value distributions of the CPTs.Furthermore, BNs can integrate results of, and help inform, other agent-based models (e.g., bioenergetic modeling), geospatial analyses, and contaminant flux and loading models used for NRDAR cases.Bayesian networks may also be used to inform alternative restoration scenarios, along with economic evaluations for NRDAR injury quantification, calculation of the quantity of restoration to offset injury, and restoration cost estimation for damage determination.Given the clear potential that BNs have for natural resource injury assessment, further research, application, and targeted hands-on guidance for NRDAR practitioners are needed.
ditions and potentially affected by environmental contaminants.The model focused on the plausible effects of Integr Environ Assess Manag 2024:562-573 Published 2023 wileyonlinelibrary.com/journal/ieam Description of site locations relative to the acid mine drainage (AMD) source in this hypothetical case study in western United States (Figure1B) supporting references).Low water pH in the model correlates with reduced Pine Dipper egg production, and it also results Integr Environ Assess Manag 2024:562-573 Published 2023 DOI: 10.1002/ieam.4836

FIGURE 2
FIGURE 2 Bayesian network showing the probability of local Pine Dipper population injury from pH and sediment copper (Cu) contamination caused by an acid mine drainage (AMD) source.Green boxes are input/parent nodes, turquoise boxes denote calculated nodes, and blue boxes are response/output nodes.AA, Assessment Area

FIGURE 3 FIGURE 4
FIGURE 3Water pH (A, B) and sediment copper (Cu; C, D) from the study.Each site has a standard boxplot plus overlaid points representing measurements, and points are jittered for ease of interpretation.For pH, previous work has indicated pH lower than 4 to drastically reduce Cinclidae nest clutch sizes(Ormerod et al., 1991), so a line at pH = 4 is on the figure for reference.Sediment Cu (C, D) has a threshold effect concentration (TEC; below which harmful effects are unlikely to be observed) of 31.6 mg/kg dw and a probable effects concentration (PEC; above which harmful effects are likely to be observed) of 149

TABLE 1
Common issues that potentially impede establishing natural resource injury, and possible solutions when using Bayesian networks (BN) as a quantitative evaluation tool • Clear definitions of nodes, states, and how they are measured • Explicitly denote uncertainties and their effects on projected outcomes • Clear expression of probabilities of events, interactions, and outcomes • Calibration and validation testing • Sensitivity analysis to determine factors most affecting outcomes Note: Each row represents a common global issue in Natural Resource Damage Assessment and Restoration (NRDAR) cases with related subissues or aspects of the problem, and possible solutions offered by use of BNs.The suggested BN solutions and methods here are supported by many resources